CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-11

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A nonfree action can have $∣ X / G ∣ \neq ∣ X ∣ / ∣ G ∣$

Statement refuted

False claim. For every action of a finite group $G$ on a finite set $X$ , the number of orbits is $∣ X ∣ / ∣ G ∣$ .

Facts & Assumptions

Given: The trivial action of $G = Z / 2$ on the singleton $X = {*}$ .

[L1]

Orbit counting uses the fixed-point sum $∣ G ∣ ∣ X / G ∣ = \sum_{g \in G} ∣ X^{g} ∣$ (Cauchy-Frobenius orbit counting: $∣ G ∣ ∣ X / G ∣ = \sum_{g \in G} ∣ X^{g} ∣$ for a finite group action).

[L2]

The trivial $Z / 2$ -action on a singleton is transitive and nonfaithful (The trivial action of $Z / 2$ on a singleton is transitive but not faithful).

Counterexample

technique · direct

1.1

By [L2], the singleton is one orbit, so $∣ X / G ∣ = 1$ .

2.1

Here $∣ X ∣ = 1$ and $∣ G ∣ = 2$ , so there is no natural number $q$ with $∣ X ∣ = ∣ G ∣ q$ ; in particular the orbit count is not obtained by dividing $∣ X ∣$ by $∣ G ∣$ .

step 1.1L2algebra

3.1

Both elements of $G$ fix the unique point, so [L1] correctly gives $2 \cdot 1 = 1 + 1$ . This verifies the orbit-counting identity while refuting the naive division rule.

step 1.1step 2.1L1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

T. W. Judson, Abstract Algebra: Theory and Applications, 14.3 (standard reference, not scraped)

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